Journal of Mathematical Biology

, Volume 48, Issue 6, pp 647–671

Large amplification in stage-structured models: Arnol’d tongues revisited


DOI: 10.1007/s00285-004-0264-8

Cite this article as:
V. Greenman, J. & Benton, T. J. Math. Biol. (2004) 48: 647. doi:10.1007/s00285-004-0264-8


The coexistence of periodic and point attractors has been confirmed for a range of stage-structured discrete time models. The periodic attractor cycles have large amplitude, with the populations cycling between extremely low and surprisingly high values when compared to the equilibrium level. In this situation a stable state can be shocked by noise of sufficient strength into a state of high volatility. We found that the source of these large amplitude cycles are Arnol’d tongues, special regions of parameter space where the system exhibits periodic behaviour. Most of these tongues lie entirely in that part of parameter space where the system is unstable, but there are exceptions and these exceptions are the tongues that lead to attractor coexistence. Similarity in the geometry of Arnol’d tongues over the range of models considered might suggest that this is a common feature of stage-structured models but in the absence of proof this can only be a useful working hypothesis. The analysis shows that although large amplitude cycles might exist mathematically they might not be accessible biologically if biological constraints, such as non-negativity of population densities and vital rates, are imposed. Accessibility is found to be highly sensitive to model structure even though the mathematical structure is not. This highlights the danger of drawing biological conclusions from particular models. Having a comprehensive view of the different mechanisms by which periodic states can arise in families of discrete time models is important in the debate on whether the causes of periodicity in particular ecological systems are intrinsic, environmental or trophic. This paper is a contribution to that continuing debate.


Arnol’d tonguesStage-structured modelsPeriodicityNoiseExternal forcingLPA model

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Computing Science and MathematicsUniversity of StirlingScotland
  2. 2.School of Biological SciencesZoology BuildingUK